Hello TomS, welcome to the vintage Airstream hobby. You are wise to learn and get properly set up with a good trailer brake controller before towing.
I would recommend you get your new brake controller installed and tested by a good shop in your area. You are right about the harness under the dash. It seems like each manufacturer does it a little differently. There are techs that do this for a living and are quite good at it. Worth the cost in my view.
Your tow vehicle has to have a stout category 4 (IV) 2" square receiver under the rear bumper. Make sure it is rated for your trailer accordingly. And your tow vehicle has to have a 7 pin trailer wiring connector attached under the bumper as well. Maybe your vehicle has these items already?
Your old Airstream is likely like my 66 Airstream. It has 12" drum brakes with 12 volt
electromagnets on a lever rubbing against the drum face as the brake actuator. The more voltage the brake controller applies to the electromagnet, the stronger the magnetic force and the harder the brakes are applied. The brake controller adjusts this voltage in proportion to the applied braking force your foot applies to the tow vehicle brake pedal.
The whole idea is the trailer brakes stop all the inertia of the weight of the trailer at speed without over burdening the tow vehicle brakes.
By the way, our trailer brakes are old fashioned adjust them yourself type. I don't know when you last did a maintenance on your trailer brakes and bearings, but I recommend it be done yearly. A good brake controller is worthless without good trailer brakes.
There is a lot to learn about trailering. Airstream Forum members have a wealth of knowledge and experience. Post your questions, and I'm sure you will get the information you need.